Simplify and expand the following expression: $ \dfrac{5}{y + 5}- \dfrac{3}{y - 3}+ \dfrac{4}{y^2 + 2y - 15} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{4}{y^2 + 2y - 15} = \dfrac{4}{(y + 5)(y - 3)}$ Now we have: $ \dfrac{5}{y + 5}- \dfrac{3}{y - 3}+ \dfrac{4}{(y + 5)(y - 3)} $ The least common multiple of the denominators is: $ (y + 5)(y - 3)$ In order to get the first term over $(y + 5)(y - 3)$ , multiply by $\dfrac{y - 3}{y - 3}$ $ \dfrac{5}{y + 5} \times \dfrac{y - 3}{y - 3} = \dfrac{5(y - 3)}{(y + 5)(y - 3)} $ In order to get the second term over $(y + 5)(y - 3)$ , multiply by $\dfrac{y + 5}{y + 5}$ $ \dfrac{3}{y - 3} \times \dfrac{y + 5}{y + 5} = \dfrac{3(y + 5)}{(y + 5)(y - 3)} $ Now we have: $ \dfrac{5(y - 3)}{(y + 5)(y - 3)} - \dfrac{3(y + 5)}{(y + 5)(y - 3)} + \dfrac{4}{(y + 5)(y - 3)} $ $ = \dfrac{ 5(y - 3) - 3(y + 5) + 4} {(y + 5)(y - 3)} $ Expand: $ = \dfrac{5y - 15 - 3y - 15 + 4}{y^2 + 2y - 15} $ $ = \dfrac{2y - 26}{y^2 + 2y - 15}$